Fluid Phase Equilibria 216 (2004) 201–217
Wax phase equilibria: developing a thermodynamic
model using a systematic approach
Hong-Yan Ji, Bahman Tohidi∗ , Ali Danesh, Adrian C. Todd
Institute of Petroleum Engineering, Heriot-Watt University, Edinburgh EH14 4AS, UK
Accepted 28 May 2003
Abstract
Reservoir hydrocarbon fluids contain heavy paraffins that may form solid phases of wax at low temperatures. Problems associated with wax
formation and deposition are a major concern in production and transportation of hydrocarbon fluids. The industry has directed considerable
efforts towards generating reliable experimental data and developing thermodynamic models for estimating the wax phase boundary.
The cloud point temperature, i.e. the wax appearance temperature (WAT) is commonly measured in laboratories and traditionally used in
developing and/or validating wax models. However, the WAT is not necessarily an equilibrium point, and its value can depend on experimental
procedures. Furthermore, when determining the wax phase boundary at pipeline conditions, the common practice is to measure the wax phase
boundary at atmospheric pressure, then apply the results to real pipeline pressure conditions. However, neglecting the effect of pressure and
associated fluid thermophysical/compositional changes can lead to unreliable results.
In this paper, a new thermodynamic model for wax is proposed and validated against wax disappearance temperature (WDT) data for a
number of binary and multi-component systems. The required thermodynamic properties of pure n-paraffins are first estimated, and then a new
approach for describing wax solids, based on the UNIQUAC equation, is described. Finally, the impact of pressure on wax phase equilibria
is addressed.
The newly developed model demonstrates good reliability for describing solids behaviour in hydrocarbon systems. Furthermore, the model
is capable of predicting the amount of wax precipitated and its composition. The predictions compare well with independent experimental
data, demonstrating the reliability of the thermodynamic approach.
© 2003 Elsevier B.V. All rights reserved.
Keywords: Wax; Solid-fluid equilibria; Equation of state; Model; Paraffin; Pressure
1. Introduction
Petroleum fluids contain heavy paraffins that may form
solid wax phases at low temperatures. Problems caused by
wax precipitation, such as decreased production rates, increased power requirements, and failure of facilities, are a
major concern in the production and transportation of hydrocarbon fluids. Techniques such as thermal treatment of
pipelines, addition of chemical inhibitors, and/or pigging are
commonly used to prevent wax accumulation. The costs associated with such measures could be reduced significantly
if accurate means to predict the wax precipitation region
were available. Therefore, it is crucial to develop reliable experimental techniques and/or predictive tools for determin∗ Corresponding author. Tel.: +44-131-451-3672;
fax: +44-131-451-3127.
E-mail address: b.tohidi@hw.ac.uk (B. Tohidi).
0378-3812/$ – see front matter © 2003 Elsevier B.V. All rights reserved.
doi:10.1016/j.fluid.2003.05.011
ing wax equilibria. The industry has directed considerable
efforts towards this goal over the past few decades.
The cloud point temperature, or wax appearance temperature (WAT), where wax is first detected on cooling, is
commonly measured in laboratories. However, WAT is not
necessarily an equilibrium point, and its value depends on
test procedures. Experimental studies conducted in this laboratory show that WAT is commonly a strong function of
cooling rate; faster cooling rates often leading to a lower
measured WAT. Furthermore, the measurement of WAT is
significantly affected by the detection techniques employed.
For example, WAT measured using visual microscopy can
be 10–20 ◦ C higher than those determined using techniques
such as differential scanning calorimetry, laser-based solids
detection systems, and viscometry [1,2].
In contrast to WAT, the wax disappearance temperature (WDT) represents a true solid–liquid equilibrium
(SLE) point. Accuracy of measured WDT is dependent on
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H.-Y. Ji et al. / Fluid Phase Equilibria 216 (2004) 201–217
experimental techniques. By using a reliable experimental method, such as equilibrium step heating, the WDT
measured by different laboratories should agree within a
reasonable experimental error band.
The difference between measured WDT and WAT can be
very significant. For example, Ronningsen et al. [1], using
microscopy, determined the WDT of a North Sea crude oil
to be 28 ◦ C higher than the WAT measured using the same
technique [1]. Therefore, WDT should be used instead of
WAT for developing, tuning and validation of wax thermodynamic models. A thermodynamic model tuned and/or validated using WAT is subject to serious questions concerning
reliability. Unfortunately, the majority of existing wax models are based on WAT, and as a result, errors in the order of
20 ◦ C between predicted and real wax phase boundaries are
common [3].
The aim of this work is to develop and validate a new
wax thermodynamic model that is based upon reliable experimental WDT data, generated both in this laboratory and
reported in the literature. We first review some popular existing wax models, and then present our new wax model and its
improvements. Predictions of this newly developed model
are then validated against independent experimental data for
wax phase boundaries, as well as the amount and composition of wax precipitated under different pressure conditions.
2. Review of existing wax thermodynamic models
Several predictive thermodynamic models for wax have
been presented in the literature. Some of the more popular
models are detailed below.
2.1. Won [4,5]
In 1986, Won presented early efforts to use a thermodynamic model for predicting wax phase boundaries [4].
The Soave–Redlich–Kwong (SRK) equation of state (EoS)
was used for vapour–liquid equilibrium (VLE) calculations.
A modified regular solution approach was employed for
solid–liquid equilibrium (SLE) calculations, where activity coefficients were calculated using solubility parameters
of individual components. The critical temperature, critical
pressure and acentric factor were estimated using correlations suggested by Spencer and Daubert [6], Lydersen [7],
and Lee–Kesler [8], respectively. The fusion temperature and
heat of fusion were correlated to molecular weight using experimental data predominantly for pure n-paraffins with odd
carbon numbers.
In 1989, Won modified the model by using an approach
that combined the modified regular solution with the equation of Flory–Huggins [9–12] for calculating activity coefficients in the liquid phase [5].
The wax model proposed by Won [4,5] was validated
against cloud point temperatures measured for synthetic fuels, diesel fuels, and North Sea gas condensates. Many other
researchers adopted the model suggested by Won [4,5],
sometimes without any modification, when developing their
own model.
However, there are several major shortcomings in the
model proposed by Won [4,5] that limit its capability and
reliability for predicting wax phase boundaries. Firstly, two
different approaches are applied to the liquid phase for VLE
and SLE; an EoS is used for VLE, while an activity coefficient model is applied to SLE. This leads to inconsistency in
description of the liquid phase, and very often results in convergence issues. A further problem is that the modified regular solution approach used for describing wax solids does not
vary greatly from the ideal solid solution approach, due to the
similarity of the solubility parameters for n-paraffins. Both
these approaches lead to overestimation of wax phase boundary temperatures. In addition, the model cannot provide reliable predictions of wax phase boundaries at high-pressure
conditions, as the effect of pressure on wax equilibria is
ignored. Finally the model is based on WAT data, which,
as discussed, are not reliable for tuning and/or validating a
model.
2.2. Hansen et al. [13]
In 1988, Hansen et al. [13] presented a wax model that
uses the SRK EoS for VLE calculations, with the ideal solid
solution approach applied to the solid phase, and a polymer
solution approach applied to the liquid phase for SLE. As
parameters required in the polymer solution approach were
determined by fitting to the measured cloud point temperatures (WAT) for 13 North Sea crude oils, it was not surprising that predicted WATs were in good agreement with
measured WAT data for the same North Sea crude oils.
The model proposed by Hansen et al. [13] has similar limitations to that of Won [4,5]. Furthermore, the polymer solution approach used by authors leads to activity coefficients
in the liquid phase of the order of 10−10 [14,15], which does
not correspond with reality [15].
2.3. Pedersen et al. [14,15]
In 1991, Pedersen et al. [14] presented a wax model based
on modifications to the approach of Won [4]. A modified
regular solution approach was applied to both the liquid and
solid phases. Fusion properties and heat capacities for pure
compounds were tuned to fit measured wax precipitation
data for the North Sea oils. The model was validated using
experimental WAT data for the North Sea oils.
In 1995, Pedersen [15] further modified this model, employing a cubic equation of state for consistency in description of the liquid phase for VLE and SLE calculations. The
ideal solid solution approach was applied to the solid phase.
Fusion properties were calculated using correlations suggested by Won [4].
A problem with the model of Pedersen et al. [14] is
that it uses unreliable values for fusion properties and heat
H.-Y. Ji et al. / Fluid Phase Equilibria 216 (2004) 201–217
capacity. The approaches used to describe wax solids in the
models proposed in 1991 [14] and 1995 [15] (i.e. the approaches of regular solid solution and ideal solid solution),
led to an overestimation of wax phase boundary temperatures. Again, models are flawed in that they are based upon
non-equilibrium WAT data.
2.4. Erickson et al. [16]
The model proposed by Erickson et al. in 1993 [16] was a
modification of that of Won [4]. The ideal solution approach
was applied to SLE calculations. Heat of fusion for pure
compounds was tuned against experimental SLE data for
binary mixtures. The proposed model was validated against
experimental WAT data for crude oil and condensate samples. The model proposed by Erickson et al. [16] has similar
limitations to that of Won [4].
2.5. LiraGaleana et al. [17]
In 1996, LiraGaleana et al. presented a wax thermodynamic model in which a multi-pure-solid approach was used
for description of wax solids [17]. This approach assumed
wax solids consisting of multiple solid phases, and each
solid phase was a pure compound. The PR EoS was used for
calculating fugacity in the liquid and vapour phases. Critical properties and the acentric factors were estimated using
correlations suggested by Cavett [18]. The model was validated using experimental SLE data for binaries, and measured cloud point temperatures (WAT) for the North Sea
crude oils.
Studies on crystal structure in recent years reveal that the
miscibility of n-paraffins in a solid state depends strongly
on differences in molecular sizes (i.e. carbon number). An
n-paraffin mixture with a significant carbon number difference (e.g. nC30 –nC36 ) appears to form eutectic solids [19],
whereas an n-paraffin mixture with a consecutive carbon
number distribution forms a single orthorhombic solid solution [20]. Thus assumptions of the multi-pure-solid approach are not consistent with real wax crystal behaviour.
Therefore, the model proposed by LiraGaleana et al. [17]
is of questionable reliability for systems consisting of compounds with similar molecular sizes.
2.6. Coutinho et al. [21–23]
In 1995 and 1996, Coutinho et al. evaluated several
approaches for calculating activity coefficients in SLE, including the Flory–Huggins, Universal Functional Group
Activity Coefficient (UNIFAC), Flory free-volume, and
entropic free-volume. In 1998, Coutinho [23] presented a
wax thermodynamic model which used a combined UNIFAC and Flory free-volume approach to describe the liquid
phase, with the universal quasi-chemical (UNIQUAC) equation being used to describe wax solids. In this UNIQUAC
approach, the characteristic energy, uij , for calculating the
203
adjustable binary parameter, τ ij , with Eq. (14), is expressed
using λij and λii as below:
uij =
z
(λij − λii )
2qi
where qi is the external surface area parameter of pure
compounds. λii is calculated using enthalpy of sublimation
for component i as follows:
2
λii = − (Hsubl,i − RT)
z
where z is the co-ordination number (set to 6 by the authors). λij is given by λkk as below, where k designates the
smaller n-alkane of the pair ij:
λij = λkk
Using the above correlations, when k is i, uij is zero, and
uji has a nonzero value. The model proposed by Coutinho
[23] was validated using experimental data for the amount
and composition of wax precipitated for mixtures.
In 2000, Pauly et al. [24] modified the model of Coutinho
[23] by using SRK EoS–GE for description of liquid and
vapour phases. GE was obtained using a modified UNIFAC
equation. Critical properties were estimated using correlations proposed by Twu [25]. The Poynting correction term
was used to extend the model to high-pressure conditions.
Partial molar volumes required for calculating the Poynting correction were estimated in accordance with crystallographic studies of n-paraffin solids. The model was validated
using experimental WDT data for n-paraffin mixtures.
Hydrocarbon solids present a positive deviation from the
ideal solid solution, as shown later in this work. This positive
deviation can be described using the UNIQUAC equation.
The accuracy of this equation depends on parameters defined
for the equation. Continho used thermodynamic properties to
calculate binary parameters. Examining the model proposed
by Coutinho [23] (as presented later in this work) shows
that Coutinho’s UNIQUAC approach lacks reliability for
mixtures containing molecules of similar sizes.
2.7. New wax model proposal
Produced reservoir hydrocarbon fluids at pipeline conditions commonly consist of liquid and vapour phases.
Vapour–liquid equilibrium is commonly calculated using
a cubic equation of state. In order to ensure consistency
in description of the liquid phase, the wax thermodynamic
model must also use a cubic equation of state for calculating fugacity in the liquid phase for SLE. SRK EoS and PR
EoS are popular for calculating fugacities in vapour–liquid
systems. These EoS are therefore suitable choices for description of fluid phases in the wax model.
When describing a compound using an equation of state,
values for critical temperature, critical pressure, and acentric factor are required. Reliable experimental data for these
parameters are available for n-paraffins up to C20 [26,27].
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However, it is almost impossible to measure directly critical properties for longer-chain n-paraffins due to thermal
decomposition at high temperatures. As a result, different
correlations have been suggested in the literature for estimation of critical properties and acentric factors for these
compounds. However, estimated values using these correlations can differ considerably, potentially having a negative
effect on the reliability of predicted wax equilibria.
Considering the limitations of existing wax models as
discussed, we propose here a new, more reliable model,
Heriot-Watt WAX (HWWAX). Firstly, we present improved
correlations for the estimation of fusion properties. Then,
the most suitable correlations for calculating Tc and Pc are
selected from those in the literature, while new correlations for calculating coefficients of temperature dependency
functions for attraction terms in the SRK EoS and PR EoS
are suggested for improving long chain n-paraffin fugacity
calculations. Following this, a new approach for describing
wax solids is developed. Finally, the effect of pressure on
wax phase equilibria is addressed, and a method is proposed
for extension of the developed wax model to high-pressure
conditions based on measurements made at atmospheric
pressure.
3. Thermodynamic modelling of wax phase equilibria
To calculate solid–liquid equilibria, it is necessary to fulfil
equality of fugacity for each component in both the solid
and liquid phases. The fugacity of component i in the liquid
phase is calculated using a cubic equation of state, expressed
as follows:
fiL = xi PϕiL
(1)
fiL
is the fugacity of component i in the liquid phase,
where
ϕiL is the fugacity coefficient, which can be calculated from
an EoS, xi is the mole fraction of the component, and P is
the system pressure.
The fugacity (fiS ) of component i in the solid phase at
system conditions is related to pure solid fugacity (fiOS ) at
a reference pressure (PO ), based on solid solution theory, as
expressed in Eq. (2):
P v̄S
i
S
S OS
dP
(2)
fi = si γi f exp
PO RT
where si and γiS are the component mole fraction and activity
coefficient in the solid phase, respectively.
fiOS is calculated from pure liquid fugacity at the same
temperature. As some n-paraffins exhibit solid–solid transitions before melting, the fugacity ratio of subcooled liquid
versus solid can be calculated using Eq. (3):
f OL
Htr,i
Hf,i
T
T
ln iOS =
+
1−
1−
RT
Ttr,i
RT
Tf,i
fi
T
T CLS
1
1
p,i
LS
+
dT
(3)
Cp,i
dT −
RT Tf,i
R Tf,i T
where Ttr,i and Tf,i are the solid–solid transition temperature
and fusion temperature of component i, respectively. Htr,i
and Hf,i are latent heats of the solid–solid and solid–liquid
LS is the heat capacity differtransitions, respectively. Cp,i
ence between liquid and solid (heat capacity differences between the two solid forms are ignored).
LS
L
S
Cp,i
= Cp,i
− Cp,i
(4)
The pure liquid fugacity of compound i (fiOL in Eq. (3))
can be calculated using a cubic equation of state.
4. Fusion properties and heat capacity
As shown in Eq. (3), fusion properties and heat capacity
are required for calculating the fugacity of solids. Accuracy
of values for these properties is vital for developing a reliable
wax model. Experimental data show that fusion properties
and heat capacities for n-paraffins are dependent not only on
the carbon chain length, but also on whether carbon numbers
are odd or even. This has been considered in the HWWAX
model when developing correlations for fusion properties
and heat capacity.
In the HWWAX model, properties for both solid–solid
(Ttr and Htr ) and solid–liquid (Tf and Hf ) transitions
(at 0.1 MPa) have been regressed into correlations using
available experimental data for pure n-paraffins up to C70
[28,29]. A third-order polynomial function has been developed to represent transition temperature as a function of carbon number, acknowledging the difference between odd and
even carbon numbers. Latent heats for transitions have been
correlated with the product of molecular weight and fusion
temperature using a linear function.
Several correlations have been developed to calculate
heat capacity for solid n-paraffins of odd or even carbon
numbers (as a function of temperature and carbon number)
using available experimental data [30–32]. A single correlation as a function of temperature and carbon number has
been developed for calculating the heat capacity of liquid
n-paraffins. Correlations for both fusion properties and heat
capacity are detailed in Appendix A of this paper.
5. Improving equations of state for calculating long
chain paraffin fugacities
Five sets of empirical correlations reported in the literature
and widely applied to hydrocarbons for estimating Tc and
Pc , have been evaluated. These include the correlations of
Ambrose [33], Twu [25], Teja et al. [34], Constantinou and
Gani [35], Riazi and Al-Sahhaf [36].
The critical volume is calculated according to:
Zc RTc
Vc =
(5)
Pc
where the critical compressibility factor (Zc ) is a constant in
both SRK EoS and PR EoS.
H.-Y. Ji et al. / Fluid Phase Equilibria 216 (2004) 201–217
Acentric factors (ω) for n-paraffins heavier than C20 are
calculated using the correlation proposed by Lee–Kesler [8].
In the original SRK EoS and PR EoS, a second-order polynomial function based on the data of pure compounds up to
C10 was used for correlating m with ω, which was required
for calculating the attraction term in the equation of state.
Direct extrapolation of m versus ω functions far beyond the
acentric factor range in which those correlations were generated can impair the reliability of the equation of state. In
this work, the m versus ω functions for SRK EoS and PR
EoS, are extended to long chain n-paraffins using optimised
m values in conjunction with calculated ω values.
The objective function for the optimisation procedure is
as follows:
F=
whilst the SLE binaries consist of a low molecular weight
n-paraffin (C5 –C12 ) and a long chain n-paraffin (C22 –C36 )
[41–46]. In the latter case, due to the large differences in the
molecular sizes of the hydrocarbons, a pure solid may form.
Therefore, the multi-pure-solid approach for describing wax
solids can be used for the optimisation of m.
Binary interaction parameters are then optimised by
matching the experimental SLE data for binaries [41–46],
using Tc and Pc calculated with the most suitable empirical
correlations. When optimising binary interaction parameters, the multi-pure-solid approach for describing wax solids
has been used for the same reason detailed in the above.
Values of critical temperature, critical pressure and the
new m versus ω functions suitable for paraffins from C1 up
to at least C36 , as well as binary interaction parameters, have
been determined for SRK and PR EoS, as presented below:
1
n1 + n 2
n
n2
1
Pb,exp − Pb,cal
WDTexp − WDTcal
+
×
P
WDT
i=1
b,exp
205
5.1. SRK EoS
exp
i=1
When using SRK EoS, correlations suggested by Riazi
and Al-Sahhaf [36] are suitable for calculating critical temperature and pressure, leading to consistency between optimised m values for n-paraffins above C20 and below C20 , as
shown in Fig. 1. Using the above data, a fourth order polynomial function for correlating m with ω in SRK EoS was
developed, as presented in Eq. (7).
(6)
where Pb is the bubble point pressure and WDT is the wax
disappearance temperature for binaries. The subscript “exp”
designates experimental data, and “cal” designates calculated data. n1 and n2 are the number for Pb data and WDT
data, respectively.
The m values for long chain n-paraffins have been optimised using estimated data of Tc and Pc with each set of
empirical correlations. According to the consistency test of
optimised m values, the most suitable empirical correlations
for estimating Tc and Pc have been selected. Both VLE and
SLE binary data were used for optimising m values, with
binary interaction parameters set to zero. The VLE binaries
consist of C2 or C3 and a n-paraffin heavier than C20 [37–40],
m = 0.4806 + 1.7137ω − 0.9207ω2
+ 0.9620ω3 − 0.2595ω4
(7)
Comparison of the m versus ω plot using Eq. (7) with that
using the original SRK second order polynomial function is
also shown in Fig. 1. As shown in Fig. 2, binary interaction parameters optimised using SLE data for binaries are
5.0
Cn < C20: the original SRK 2rd order polynomial function
Cn > C20: the original SRK 2rd order polynomial function
Cn > C20: optimized m values with SRK EoS, this work
C1 - C60: the 4th order polynomial function for SRK EoS, this work
4.5
4.0
m value
3.5
C36
3.0
C32
2.5
C22
C24
C20
2.0
1.5
1.0
0.5
0.0
0.0
0.2
0.4
0.6
0.8
1.0
1.2
1.4
1.6
1.8
2.0
acentric factor
Fig. 1. Relationship between m values and acentric factor (ω) in SRK EoS (Tc and Pc calculated using correlations suggested by Riazi and Al-Sahhaf
[36] and ω calculated using the correlation suggested by Lee–Kesler [8] for n-paraffins above C20 ).
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H.-Y. Ji et al. / Fluid Phase Equilibria 216 (2004) 201–217
0.00
Binary interaction parameter
-0.01
-0.02
-0.03
-0.04
-0.05
The heavier component: C22
-0.06
The heavier component: C24
-0.07
The heavier component: C28
-0.08
The heavier component: C32
The heavier component: C36
-0.09
-0.10
4
5
6
7
8
9
10
11
12
13
14
Cn of the lighter component in binary systems
Fig. 2. Binary interaction parameter optimised using SRK EoS and SLE data in binaries (Cn : carbon number).
developing the following function for correlating m with ω:
almost constant and independent of the combination (pairing) of compounds when using SRK EoS. The value of binary interaction parameter is approximately −0.02 for all
the binaries investigated.
m = 0.3748 + 1.5932ω − 0.5706ω2
+ 0.3968ω3 − 0.092ω4
(8)
A comparison of the m versus ω plot using Eq. (8) with
that using the original PR second-order polynomial function
is shown in Fig. 3. As shown in Fig. 4, binary interaction
parameters optimised using the SLE data for binaries are
almost independent of compound combinations, with the
average close to −0.024 when PR EoS is applied to the
calculation of fugacity in the liquid phase.
5.2. PR EoS
When using PR EoS, the correlations suggested by Twu
[25] are suitable for calculating critical temperature and critical pressure for n-paraffins above C20 . As for SRK EoS,
the m values optimised for n-paraffins above C20 and those
for n-paraffins below C20 (shown in Fig. 3) were used in
5.0
Cn < C20: the original PR 2rd order polynomial function
4.5
Cn > C20: the original PR 2rd order polynomial function
4.0
Cn > C20: optimized m values with PR EoS, this work
m value
3.5
C1 - C60: the 4th order polynomial function for PR EoS, this work
3.0
2.5
2.0
C20
C22
C24
C32
C36
1.5
1.0
0.5
0.0
0.0
0.2
0.4
0.6
0.8
1.0
1.2
1.4
1.6
1.8
2.0
acentric factor
Fig. 3. Relationship between m values and acentric factor (ω) in PR EoS (Tc and Pc calculated using correlations suggested by Twu [25] and ω calculated
using the correlation suggested by Lee–Kesler [8] for n-paraffins above C20 ).
H.-Y. Ji et al. / Fluid Phase Equilibria 216 (2004) 201–217
207
0.00
Binary interaction parameter
-0.01
-0.02
-0.03
-0.04
-0.05
The heavier component: C22
-0.06
The heavier component: C24
-0.07
The heavier component: C28
The heavier component: C32
-0.08
The heavier component: C36
-0.09
-0.10
4
5
6
7
8
9
10
11
12
13
14
Cn of the lighter component in binary systems
Fig. 4. Binary interaction parameter optimised using PR EoS and SLE data for binaries (Cn : carbon number).
6. Describing wax solids
The UNIQUAC equation with parameters determined in
this work is used for describing wax solids, i.e. for calculating activity coefficients in the solid phase. The general
UNIQUAC equation in terms of molar excess Gibbs energy
is given as:
n
n
gE
z
ϑi
θi
si ln
qi si ln
+
=
RT
si
2
ϑi
i=1
i=1
n
n
− qi si ln θj τji
i=1
As shown in Eq. (9), for each pair of compounds, there
are two adjustable parameters, τ ij and τ ji . These are given in
terms of characteristic energies uij and uji . The general
formulation for calculating τ ij is:
uij
τij = exp −
(14)
RT
The characteristic energy, uij , is correlated with Cn,ij ,
the difference in carbon numbers for a pair of compounds:
uij = a × Cn,ij
(9)
j=1
with
where a is a constant determined as 11 in this work using
the experimental WDT data generated in this laboratory for
C16 –C18 , C16 –C20 and C15 –C19 binaries.
This work further considers that:
ϑi =
ri si
n
j=1 rj sj
(10)
uji = uij
θi =
qi si
n
j=1 qj sj
(11)
7. Modelling high pressure conditions
where gE is the molar excess Gibbs energy. ri and qi are
molecular structure parameters of pure compounds, which
depend on molecular sizes and external surface areas. z is
the coordination number (6 ≤ z ≤ 12).
The coordination number, z, is set to 10 here, according to
the value suggested by Abrams and Prausnitz [47]. Based on
the n-paraffin structure parameters provided in the literature
[47,48], the following correlations have been developed for
calculating ri and qi .
ri = 0.675Cn,i + 0.4483
(12)
qi = 0.540Cn,i + 0.6200
(13)
where Cn,i is the carbon number for compound i.
(15)
(16)
The exponential term in Eq. (2), referred as the Poynting
correction term, takes into account the effect of the difference between the operating pressure (P) and the reference
pressure (PO ). In this work, the reference pressure is set
to be the operating pressure, where the Poynting correcting
term becomes unity. To calculate the pure solid fugacity at
the reference pressure (fi OS ), fusion properties of pure compounds have to be those at the operating pressure conditions
(i.e. the reference pressure in this work).
Based on experimental measurements for pure n-paraffins
[49,50], the following generalized correlation is proposed for
calculating fusion temperatures at increased pressure conditions.
Tf(P) = Tf(P=0.1 MPa) + 0.2 × (P − 0.1)
(17)
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H.-Y. Ji et al. / Fluid Phase Equilibria 216 (2004) 201–217
where Tf(P=0.1 MPa) and Tf(P) are the fusion temperature of
pure compounds at 0.1 MPa and the operating pressure (P)
in MPa, respectively.
8. Results and discussion
8.1. Validating correlations of fusion properties and heat
capacity
Independent experimental WDT data for C6 –C16 and
C6 –C17 binaries, generated in this laboratory, have been
used for validating the suggested correlations for calculating fusion properties and heat capacity in this work.
Critical properties and acentric factors for the compounds
in these binaries are reliable and reported in the literature
such as CRC [26]. Furthermore, pure solids may form in
these binaries due to the significant differences in molecular
sizes. Hence the accuracy of predicted WDT for these binaries largely depends on the reliability of values for fusion
properties and heat capacity.
As shown in Fig. 5, WDT predictions using the model developed in this work (HWWAX) are in good agreement with
our experimental data, demonstrating the reliability of the
fusion properties and heat capacity correlations. It may be
inferred that the inclusion of a solid–solid transition in SLE
calculations and acknowledging the differences between odd
and even carbon numbers of n-paraffins, have improved the
reliability of HWWAX predictions.
8.2. Comparing wax solid models
Experimental data for binaries and ternaries, generated
in this laboratory or reported in the literature, have been
used for comparing several approaches for the description
of wax solids. These include the ideal solid solution, the
multi-pure-solid, Coutinho’s UNIQUAC, and finally the
HWWAX UNIQUAC approaches. Compounds in the binaries and ternaries are lighter than C20 . Their critical properties and acentric factors have been measured experimentally
and reported in reference handbooks. Hence, the reliability of WDT predictions for these mixtures depends on the
thermodynamic model used for describing wax solids.
8.2.1. Binaries
Experimental WDT data have been generated for C16 –
C18 , C16 –C20 and C15 –C19 binaries in this laboratory. As
shown in Figs. 6–8, the predictions using the ideal solid solution approach generally overestimate WDTs for all the binaries investigated. This suggests that the solid solution for
these binaries is non-ideal, and the deviation from ideal is
positive. However, as shown in Figs. 7 and 8, the multi-puresolid approach is suitable for mixtures with components having significant differences in chain-length (e.g. C16 –C20 and
C15 –C19 binaries), whereas, it is not satisfactory for predicting WDT for n-paraffins when molecular size differences
are small (e.g. C16 –C18 binary). Coutinho’s UNIQUAC approach also shows limitations in the description of the nonideality of solid solutions formed by n-paraffins with similar
carbon numbers (e.g. C16 –C18 binary). Experimental SLE
data for C16 –C18 , C16 –C20 and C15 –C19 binaries have been
used in the optimisation process for determining the constant in Eq. (15) when developing the HWWAX UNIQAUC
approach for describing wax solids. Calculated wax phase
boundaries using HWWAX are in good agreement with the
experimental measurements, as shown in Figs. 6–8.
Experimental WDT data for the binary C17 –C19 system have been reported in the literature [51]. As shown in
Fig. 9, independent WDT predictions using HWWAX are in
excellent agreement with the measurements, demonstrating
280
275
270
WDT/K
265
260
255
C6-C16: exp. data, this laboratory
250
C6-C16: HWWAX predictions
245
C6-C16: predictions without classifying odd/even Cn
240
C6-C17: exp. data, this laboratory
C6-C17: HWWAX predictions
235
C6-C17: predictions without inclusion of S-S transition
230
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
C16 or C17 mole fraction
Fig. 5. Experimental (this laboratory) and predicted WDTs for C6 –C16 and C6 –C17 binaries.
H.-Y. Ji et al. / Fluid Phase Equilibria 216 (2004) 201–217
Exp. data: this laboratory
Calculations: HWWAX
Predictions: the ideal solid solution
Predictions: the multi-pure-solid approach
Predictions: Countinho's UNIQUAC approach
305
300
WDT/K
209
295
290
285
0
0.2
0.4
0.6
0.8
1
C18 mole fraction
Fig. 6. Comparison of experimental WDT data (this laboratory) for C16 –C18 binaries with model predictions using several approaches.
the reliability of the approach used for describing wax
solids.
8.3. The wax phase boundary for multi-component
mixtures: effect of pressure
8.2.2. Ternaries
Experimental WDT data for C14 –C15 –C16 and C18 –C19 –
C20 ternaries have been reported in the literature [52]. WDT
data for C6 –C16 –C17 ternaries have been generated in this
laboratory. As shown in Tables 1–3, HWWAX predictions
show very good agreement with experimental data over a
wide range of compositional distributions. A further observation is that the ideal solid solution approach generally
overestimates WDTs, while the multi-pure-solid approach
and Coutinho’s UNIQUAC approach shows significant deviations in some ternary systems.
Temperature is the major factor affecting wax precipitation. However, the impact of pressure on wax formation is
also significant. Although wax may not form within operating temperatures at atmospheric pressure (the condition
under which WATs and WDTs are often measured), wax
precipitation may occur at higher pressures. Therefore, it is
important for a wax model to have the ability to provide
reliable prediction of wax phase equilibria at high-pressure
conditions.
Pressure affects wax precipitation in two opposing ways.
Firstly, the solubility of gases in the liquid phase increases
312
307
WDT/K
302
297
292
Exp. data: this laboratory
Calculations: HWWAX
Predictions: the ideal solid solution
Predictions: the mullti-pure-solid approach
Predictions: Coutinho's UNIQUAC approach
287
282
0
0.2
0.4
0.6
0.8
1
C20 mole fraction
Fig. 7. Comparison of experimental WDT data (this laboratory) for C16 –C20 binaries with model predictions using several approaches.
210
H.-Y. Ji et al. / Fluid Phase Equilibria 216 (2004) 201–217
306
301
WDT/K
296
291
286
Exp. data: this laboratory
281
Calculations: HWWAX
Predictions: the ideal solid solution
276
Predictions: the multi-pure-solid approach
Predictions: Coutinho's UNIQUAC approach
271
0
0.2
0.4
0.6
0.8
1
C19 mole fraction
Fig. 8. Comparison of experimental WDT data (this laboratory) for C15 –C19 binaries with model predictions using several approaches.
boundary may be shifted with respect to the atmospheric
pressure condition. Obviously, for a system above its bubble
point, only the second factor will influence the wax phase
boundary.
In order to evaluate the reliability of HWWAX predictions, WDTs were measured for three multi-component mixtures at different pressures in this laboratory. Compositions
for the mixtures are listed in Table 4.
As shown in Fig. 10, the predicted wax phase boundaries
using HWWAX are in good agreement with the independent
experimental data measured in our laboratory. The pressure
impact on WDT is obvious. As the pressure increases to
40 MPa, the WDT increases by approximately 8 K.
with an increase in the system pressure at conditions below
the bubble point. This extra dissolved gas leads to a lower
mole fraction of heavy compounds in the liquid phase, which
may reduce the tendency for wax formation. On the other
hand, as pressure increases, the solidification temperature
of pure compounds increases, and the wax phase boundary
may shift to a higher temperature at any given pressure. The
impact of pressure on fugacity of condensed phases away
from the critical point is not very significant in comparison
with the above two effects.
In a gas–liquid system, both factors mentioned above may
affect the wax phase boundary. Under high-pressure conditions, depending on the dominant factor, the wax phase
306
WDT/K
301
296
291
Exp: Robles et al. (1996)
Predictions: HWWAX
Predictions: the ideal solid solution approach
Predictions: the multi-pure-solid approach
Predictions: Coutinho's UNIQUAC approach
286
281
0
0.2
0.4
0.6
0.8
1
C19 mole fraction
Fig. 9. Comparison of experimental WDT data [51] for C17 –C19 binaries with model predictions using several approaches.
H.-Y. Ji et al. / Fluid Phase Equilibria 216 (2004) 201–217
211
Table 1
Experimental WDT data [52] and model predictions for C14 –C15 –C16 ternary, at 0.1 MPa
Experimental data
Predictions and deviations (Dev)
Mole fraction
C14
C15
C16
0.06
0.14
0.17
0.24
0.21
0.27
0.37
0.32
0.43
0.57
0.73
0.57
0.23
0.06
0.33
0.56
0.66
0.05
0.24
0.33
0.17
0.14
0.37
0.63
0.77
0.43
0.23
0.07
0.58
0.44
0.24
0.26
0.13
Exp. WDT
(K)
283
285
286
282
281
280
283
282
279
278
276
Ideal solid solution
Multi-pure-solid
Countinho’s UNIQUAC
HWWAX
WDT (K)
Dev (K)
WDT (K)
Dev (K)
WDT (K)
Dev (K)
WDT (K)
Dev (K)
287
288
289
286
285
283
287
286
284
284
282
4
3
3
4
4
3
4
4
5
6
6
277
284
287
279
272
275
283
280
272
273
274
−6
−1
1
−3
−10
−5
0
−2
−7
−5
−2
280
285
287
281
276
278
284
281
276
276
275
−3
0
1
−1
−5
−2
1
−1
−3
−2
−1
285
287
288
284
283
281
285
284
281
280
279
2
2
2
2
2
1
2
2
2
2
3
Table 2
Experimental WDT data [52] and model predictions for C18 –C19 –C20 ternary, at 0.1 MPa
Experimental data
Predictions and deviations (Dev)
Mole fraction
C18
C19
C20
0.02
0.05
0.05
0.1
0.1
0.1
0.14
0.15
0.2
0.2
0.26
0.33
0.4
0.43
0.48
0.6
0.79
0.9
0.02
0.05
0.9
0.1
0.4
0.55
0.73
0.15
0.2
0.6
0.26
0.33
0.1
0.43
0.15
0.2
0.11
0.05
0.96
0.9
0.05
0.8
0.5
0.35
0.13
0.7
0.6
0.2
0.48
0.34
0.5
0.14
0.37
0.2
0.1
0.05
Exp. WDT
(K)
Ideal solid solution
Multi-pure-solid
Countinho’s UNIQUAC
HWWAX
WDT (K)
Dev (K)
WDT (K)
Dev (K)
WDT (K)
Dev (K)
WDT (K)
Dev (K)
309
309
305
308
306
306
304
307
306
305
306
304
305
303
304
302
301
301
310
309
305
309
308
307
306
308
308
306
307
306
307
304
306
304
303
302
1
0
0
1
2
1
2
1
2
1
1
2
2
1
2
2
2
1
309
309
303
307
302
298
300
306
304
296
301
298
302
292
299
295
298
299
0
−1
−2
−1
−4
−8
−4
−1
−2
−9
−5
−7
−3
−11
−6
−8
−3
−2
309
309
304
307
303
300
301
306
305
296
302
299
303
297
300
295
299
300
0
0
−1
−1
−3
−6
−3
−1
−1
−10
−4
−5
−2
−6
−4
−7
−2
−1
309
309
305
308
306
305
304
307
306
304
305
303
304
302
303
301
300
301
0
0
0
0
0
−1
0
0
0
−1
−1
−1
−1
−1
−1
−1
−1
−1
efficient and economical use of techniques to prevent wax
accumulation. Experimental data for the composition and
amount of wax precipitated for a multi-component mixture
under several temperature and pressure conditions have been
8.4. Amount of wax precipitated and its composition
The ability to predict the amount and composition of wax
formed at given conditions could be extremely useful for
Table 3
Experimental WDT data (Heriot-Watt) and model predictions for C6 –C16 –C17 ternary, at 0.1 MPa
Experimental data
Predictions and deviations (Dev)
Mole fraction
C6
C16
C17
0.911
0.905
0.794
0.048
0.04
0.156
0.041
0.055
0.051
Exp. WDT
(K)
Ideal solid solution
Multi-pure-solid
Countinho’s UNIQUAC
HWWAX
WDT (K)
Dev (K)
WDT (K)
Dev (K)
WDT (K)
Dev (K)
WDT (K)
Dev (K)
261
263
271
265
266
273
4
3
2
254
257
267
−7
−6
−4
258
260
268
−3
−3
−3
261
262
270
0
−1
−1
212
H.-Y. Ji et al. / Fluid Phase Equilibria 216 (2004) 201–217
Table 4
Compositions (mol%) for mixtures A, B and C (Heriot-Watt)
reported in the literature [53]. The mixture consisted of consecutive normal alkanes from C6 to C36 , with decreasing
concentration as a function of increasing in chain-length,
representing a highly simplified crude oil system.
Components
A
B
C
51.04
44.49
1.59
2.12
0.33
–
–
0.17
–
0.19
–
–
0.08
–
80.04
–
–
–
6.43
4.39
2.99
2.06
2.34
1.41
0.34
–
47.45
37.76
–
6.44
2.4
3.23
1.81
0.22
0.3
–
0.21
0.18
–
8.4.1. Amount of wax
The amounts of wax (mass% of the feed) predicted using different models (i.e. HWWAX, Coutinho’s UNIQUAC
approach, and the ideal solid solution approach) are compared against independent experimental data in Fig. 11. It
is obvious that the ideal solid solution approach overestimates the solid amounts over the whole temperature range
(i.e. 256–290 K). Predictions of the HWWAX model are in
good agreement with the experimental data. As expected,
the amount of wax deposition increases with a decrease in
the system temperature.
50
45
40
P/MPa
35
30
25
A: exp. data, this laboratory
20
A: HWWAX predictions
15
B: exp. data, this laboratory
B: HWWAX predictions
10
C: exp. data, this laboratory
5
0
285
C: HWWAX predictions
290
295
300
305
310
315
320
325
330
T/K
Fig. 10. Measured (this laboratory) and predicted (using HWWAX) WDTs for mixtures A, B and C at different pressure conditions.
40
0.1MPa, Exp. data: Pauly et al. (2001)
0.1 MPa, Predictions: HWWAX
0.1 MPa, Predictions: Ideal solid solution approach
0.1 MPa, Predictions: Coutinho's UNIQUAC approach
50MPa, Exp. data: Pauly et al. (2001)
50MPa, Predictions: HWWAX
35
Solid depost/mass%
C7
C10
C13
C16
C18
C20
C21
C22
C23
C24
C28
C30
C36
30
25
20
15
10
5
0
250
260
270
280
290
300
T/K
Fig. 11. Measured wax amounts (mass%) [53] and predictions using different wax models at 0.1 MPa.
H.-Y. Ji et al. / Fluid Phase Equilibria 216 (2004) 201–217
213
30
Solid composition/mass%
Exp. data: Pauly et al. (2001)
25
Predictions: HWWAX
Predictions: Ideal solid solution approach
20
Predictions: Coutinho's UNIQUAC approach
15
10
5
0
5
10
15
20
25
30
35
40
Carbon numer
Fig. 12. Measured [53] and predicted wax compositions using different wax models at 290.2 K and 0.1 MPa.
deviation for the concentration of heavy hydrocarbons in the
wax. Predictions of wax composition using HWWAX are in
close agreement with experimental data. The predicted wax
deposition at 290 K and 0.1 MPa, predominantly consists of
paraffins heavier than C25 , which are in good agreement
with the measured data.
Fig. 13 shows the effect of temperature on the composition of wax precipitates (at 0.1 MPa). As temperature reduces, more light hydrocarbons take part in wax formation.
The predictions of HWWAX are in good agreement with
experimental data at different temperatures.
Fig. 14 shows the impact of pressure on the composition
of the wax phase. An increase in pressure has a similar
impact as a reduction in temperature; both cause more light
The reliability of HWWAX for prediction of the effect of
pressure on wax deposition is also demonstrated in Fig. 11.
In this case, as the pressure increases from atmospheric pressure to 50 MPa, the amount of wax precipitated increases
from 4.6 to 9.3 mass% at 273.2 K.
8.4.2. Wax composition
As shown in Fig. 12, the ideal solid solution approach
over-estimates the amount of light components in the precipitated wax. Clearly, concentrations of n-paraffins below C28
in the wax are highly overestimated when using the ideal
solid solution approach. Predictions based on the Coutinho’s
UNIQUAC approach are in reasonable agreement with experimental data. However, this approach shows an obvious
30
Solid composition/mass%
290.2K, Exp. data: Pauly et al. (2001)
290.2K, Predictions: HWWAX
25
273.2K, Exp. data: Pauly et al. (2001)
273.2K, Predictions: HWWAX
20
256.2K, Exp.data: Pauly et al. (2001)
256.2K, Predictions: HWWAX
15
10
5
0
5
10
15
20
25
30
35
40
Carbon number
Fig. 13. Measured [53] and predicted (using HWWAX) wax compositions at different temperature conditions, at 0.1 MPa.
214
H.-Y. Ji et al. / Fluid Phase Equilibria 216 (2004) 201–217
25
0.1MPa, Exp.data: Pauly et al. (2001)
Solid composition/mass%
0.1MPa, Predictions: HWWAX
20
50MPa, Exp. data: Pauly et al. (2001)
50MPa, Predictions: HWWAX
15
10
5
0
5
10
15
20
25
30
35
40
Carbon number
Fig. 14. Experimental [53] and predicted (using HWWAX) wax compositions at different pressure conditions (290.2 K).
hydrocarbons to take part in the wax phase. The predictions
using HWWAX are in good agreement with experimental
data at different pressures.
9. Conclusions
A thermodynamically consistent phase behaviour model
to predict the phase boundary, amount and composition of
wax (HWWAX) has been developed in this work. The reliability of HWWAX has been verified by comparing its predictions with independent experimental data and those of
other leading wax models.
The reliability of the model is attributed to: (1) introduction of higher-accuracy values for fusion properties and heat
capacities for n-paraffins, (2) optimisation of basic parameters for long chain n-paraffins for use in equations of state,
(3) development of a new and more reliable approach for
describing wax solids, and (4) extension of model capabilities to high-pressure conditions using the thermodynamic
properties of pure compounds.
List of symbols
a
constant
Cn
carbon number
Cp
heat capacity
f
fugacity
g
molar Gibbs energy
H
enthalpy
P
pressure
q
molecular external surface
r
molecular size
R
gas constant
s
T
u
V
x
v
z
Z
solid mole fraction
temperature
characteristic energy
volume
liquid mole fraction
molar volume
coordination number
compressibility factor
Greek
γ
ϕ
ϑ
θ
letters
variation
activity coefficient
fugacity coefficient
parameter related with molecular size
parameter related with molecular
external surface
parameter related with characteristic energy
acentric factor
τ
ω
Superscripts
E
excess
L
liquid phase
O
pure component
S
solid phase
Subscripts
b
bubble point
c
critical condition
cal
calculated data
exp experimental data
f
fusion
O
reference condition
subl sublimation
tr
solid–solid transition
H.-Y. Ji et al. / Fluid Phase Equilibria 216 (2004) 201–217
215
• Others
Acknowledgements
This work was part of a Joint Industrial project funded by
ABB Offshore Systems Ltd., the UK Department of Trade
and Industry, Petrobras, Shell UK Exploration and Production, and TOTAL, whose support is gratefully acknowledged.
Hongyan Ji wishes to thank James Watt Scholarship and the
ORS Award Scheme for financial support. The authors also
wish to thank Mr. Rod Burgass for his contributions to experimental work, and Mr. Ross Anderson for his assistance
in the revision of manuscript.
Ttr (K) = Tf
n-Paraffins with even carbon numbers:
• For C22 ≤ Cn ≤ C42
Ttr (K) = 0.0032C3n − 0.3249C2n + 12.78Cn
+ 154.19 + ln(Cn )
• Others
Ttr (K) = Tf
Appendix A
A.2. Heat of fusion and heat of solid–solid transitions
A.1. Fusion temperature and solid–solid transition
temperature
Correlations for calculating fusion temperatures and
solid–solid transition temperatures have been developed in
accordance with measured values reported in the literature
[28,29]. Differentiation between odd or even carbon numbers for n-paraffins is applied to correlations in order to
improve accuracy.
A.1.1. Fusion temperature
n-Paraffins with odd carbon numbers:
• For Cn ≤ C9
• For C9 < Cn ≤ C43
Tf (K) = 0.0122C2n − 2.0861Cn − 775.598/Cn
+ 76.2189 ln(Cn ) + 156.9
Hsum (cal mol−1 ) = 0.167MW × Tf + 432.47
• For Cn > C33
Hsum (cal mol−1 ) = 0.139MW × Tf + 3984.8
Hsum (cal mol−1 ) = 0.180MW × Tf + 522.7
• For Cn > C34
• For Cn ≤ C10
Tf (K) = −0.0998C3n + 1.0812C2n + 18.602Cn + 49.216
• For C10 < Cn ≤ C42
− 0.3458C2n
+ 14.277Cn + 137.73
• For Cn > C42
Tf (K) =
Hsum (cal mol−1 ) = 0.119MW × Tf + 672.2
• For Cn ≤ C34
414.3(Cn − 1.5)
Cn + 5.0
n-Paraffins with even carbon numbers:
Tf (K) =
• For Cn ≤ C9
n-Paraffins with even carbon numbers:
• For Cn > C43
0.0031C3n
A.2.1. Sum of heats of fusion and heats of solid–solid
transitions
n-Paraffins with odd carbon numbers:
• For C9 < Cn ≤ C33
Tf (K) = 0.3512C3n − 7.6438C2n + 72.898Cn − 73.9
Tf (K) =
The sum of heats of fusion and heats of solid–solid transitions are considered to be dependent on the fusion temperature and the molecular weight. Correlations have been
developed using data reported in the literature [28,29].
414.3(Cn − 1.5)
Cn + 5.0
A.1.2. Solid–solid transition temperatures
n-Paraffins with odd carbon numbers:
• For C9 < Cn ≤ C43
Ttr (K) = 0.0039C3n −0.4239C2n +17.28Cn −ln(Cn )+95.4
Hsum (cal mol−1 ) = 0.139MW × Tf + 3984.8
A.2.2. Heat of fusion and heat of solid–solid transitions
n-Paraffins with odd carbon number:
• For Cn ≤ C9
Hf (cal mol−1 ) = 1.0 Hsum
Htr (cal mol−1 ) = 0
• For C9 < Cn ≤ C43
Hf (cal mol−1 ) = 0.74 Hsum
Htr (cal mol−1 ) = 0.26 Hsum
216
H.-Y. Ji et al. / Fluid Phase Equilibria 216 (2004) 201–217
• For Cn > C20
• For Cn > C43
Hf (cal mol−1 ) = 1.0 Hsum
aS = (1.6964 × Cn − 22.5000) × 10−6
Htr (cal mol−1 ) = 0
bS = −(1.1670 × Cn − 19.525) × 10−3
n-Paraffins with even carbon number:
d S = −(1.5093 × Cn − 31.209) × 10
cS = (2.4703 × Cn − 39.848) × 10−1
• For Cn ≤ C20
Hf (cal mol−1 ) = 1.0 Hsum
Htr (cal mol
−1
)=0
• For C20 < Cn ≤ C42
Hf (cal mol−1 ) = 0.64 Hsum
Htr (cal mol−1 ) = 0.36 Hsum
• For Cn > C42
Hf (cal mol−1 ) = 1.0 Hsum
Htr (cal mol−1 ) = 0
A.3. Heat capacity
Correlations have been developed for calculating heat capacity as a function of temperature and carbon number using
measured data for n-paraffins [30–32]. Differentiation between odd or even carbon numbers for n-paraffins is applied
to correlations when calculating the heat capacity of solids.
A.3.1. Heat capacity for n-paraffin liquids
CpL (cal mol−1 K−1 ) = aL T + bL
with
aL = 0.01 × Cn − 0.0138
bL = 4.529 × Cn + 3.8457
A.3.2. Heat capacity for n-paraffin solids
CpS (cal mol−1 K−1 ) = aS T 3 + bS T 2 + cS T + d S
with
n-Paraffins with odd carbon number (based on data available up to C19 ):
aS = (0.3571 × Cn + 2.1667) × 10−6
bS = −(0.2014 × Cn + 0.4300) × 10−3
cS = (0.4579 × Cn + 0.8105) × 10−1
d S = −(0.0678 × Cn − 0.0580) × 10
n-Paraffins with even carbon number:
• For Cn ≤ C20
aS = (0.0929 × Cn + 4.9286) × 10−6
bS = −(0.0993 × Cn + 1.5929) × 10−3
cS = (0.3604 × Cn + 1.9115) × 10−1
d S = −(0.0459 × Cn + 0.2022) × 10
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